On the Direct Integral Decomposition in Branching Laws for Real Reductive Groups
Journal of Lie Theory, Tome 32 (2022) no. 1, pp. 191-196
Voir la notice de l'article provenant de la source Heldermann Verlag
The restriction of an irreducible unitary representation $\pi$ of a real reductive group $G$ to a reductive subgroup $H$ decomposes into a direct integral of irreducible unitary representations $\tau$ of $H$ with multiplicities $m(\pi,\tau)\in\mathbb{N}\cup\{\infty\}$. We show that on the smooth vectors of $\pi$, the direct integral is pointwise defined. This implies that $m(\pi,\tau)$ is bounded above by the dimension of the space Hom$_H(\pi^\infty|_H,\tau^\infty)$ of intertwining operators between the smooth vectors, also called \emph{symmetry breaking operators}, and provides a precise relation between these two concepts of multiplicity.
Classification :
22E45, 22E46
Mots-clés : Real reductive groups, unitary representations, branching laws, direct integral, pointwise defined, smooth vectors, symmetry breaking operators
Mots-clés : Real reductive groups, unitary representations, branching laws, direct integral, pointwise defined, smooth vectors, symmetry breaking operators
Affiliations des auteurs :
Jan Frahm 1
Jan Frahm. On the Direct Integral Decomposition in Branching Laws for Real Reductive Groups. Journal of Lie Theory, Tome 32 (2022) no. 1, pp. 191-196. http://geodesic.mathdoc.fr/item/JOLT_2022_32_1_a9/
@article{JOLT_2022_32_1_a9,
author = {Jan Frahm},
title = {On the {Direct} {Integral} {Decomposition} in {Branching} {Laws} for {Real} {Reductive} {Groups}},
journal = {Journal of Lie Theory},
pages = {191--196},
year = {2022},
volume = {32},
number = {1},
url = {http://geodesic.mathdoc.fr/item/JOLT_2022_32_1_a9/}
}